Distribution of Farey fractions in residue classes and Lang–Trotter conjectures on average

Abstract

We prove that the set of Farey fractions of order T T , that is, the set { α / β ∈ Q : gcd ⁡ ( α , β ) = 1 , 1 ≤ α , β ≤ T } \{\alpha /\beta \in \mathbb {Q}\ : \ \operatorname {gcd}(\alpha , \beta ) = 1, \ 1 \le \alpha , \beta \le T\} , is uniformly distributed in residue classes modulo a prime p p provided T ≥ p 1 / 2 + ε T \ge p^{1/2 +\varepsilon } for any fixed ε > 0 \varepsilon >0 . We apply this to obtain upper bounds for the Lang–Trotter conjectures on Frobenius traces and Frobenius fields “on average” over a one-parametric family of elliptic curves.